Global and local dynamics in correlated systems T. Di Matteo, T. Aste, F. Pozzi T. Di Matteo, T. Aste, F. Pozzi Department of Applied Mathematics M. Tumminello and R. N. Mantegna Giulia Rotundo

Characterization and Visualization of financial markets by means of Hyperbolic networks New correlation filtering procedure Planar Maximally Filtered Graph (PMFG) An application to interest rates, 100 stocks of US equity market, 300 stocks NYSE Topological properties : degree, betweenness, average length of shortest paths at different time horizons (returns) Dynamical filtered graphs at different time windows Brief overview

A ustralian R esearch C ouncil Project: “The architecture of networks: Characterization and Visualization of complex systems as fluctuating networks” Characterize the statistical, geometrical and topological properties of complex systems by mapping the structure of their interactions into graphs in multidimensional spaces, both Euclidean and non-Euclidean.

G. Ringel, Map Color Theorem, Springer-Verlag, Berlin, (1974) cap. 4 P. J. Gilbin, Graphs, Surfaces and Homology, Chapman and Hall, 2 nd edition (1981) G. Ringel and J. W. T. Youngs, Proc. Nat. Acad. Sci. USA 60 (1968) The embedding of K n is possible on an orientable surface S g of genus 2D hyperbolic surface Locally planarLocally planar natural hierarchynatural hierarchy characterizationcharacterization elementary moveselementary moves WHY NOT? WHY NOT? WHY SURFACES ? any  n is a sub-graph of K n and can be embedded on S g

Which SURFACES? Which SURFACES? g = 0sphere 0 non-contractible loops 1 cut g = 1torus 2 non-contractible loops 2 cuts g = 2 4 non-contractible loops 3 cuts

Planar graph g=0 K5K5 K 3,3 Kuratowski’s theorem A finite graph is planar if and only if it does not contain a subgraph that is an expansion of K 5 or K 3,3

WEIGHTS WEIGHTS The relevance of a link between two node is measured in term of a scalar quantity: the weight or the cost. weightcomplete graph Given a weight for each of the n(n-1)/2 links in the complete graph, sub-graphmaximal information constraining complexity construct a sub-graph of K n which retains maximal information (minimal weight) while constraining complexity. Construction of graph from the weights:

Fix g If and only if the resulting graph can be embedded on a surface of genus g connect two nodes n unconnected nodes T. Aste, T. Di Matteo and S. T. Hyde, Complex Networks on Hyperbolic Surfaces, Physica A 346 (2005) cond-mat/ Bottom Up complete graph K n Unfold S g* into its universal cover H 2 Embedding on S g* Top Down Edge pruning H 2 Regluing the universal cover on S g in E n Arbitrary graph on S g Glauber dynamics Local elemetary move Dynamical

Application to interest rates Eurodollar Interest Rates with maturity dates between 3 to 48 months T. Di Matteo, T. Aste, Int. J. of Theor. and Appl. Finance. 5 (2002) 107

Federal funds rate (FED) State & local bonds (SLB) Commercial Paper (CP) Finance Paper placed directly (FP) Bankers acceptances (BA) Rate on certificates of deposit (CD) Treasury securities at ‘constant maturity’ (TC) Treasury bill rates (TBA) Treasury bill secondary market rates (TBS) Treasury long-term bond yield (TC10P) Eurodollar interbank interest rates (ED) Corporate bonds Moody’s seasoned rates (AAA, BAA) Conventional mortgages rates (CM) T. Di Matteo, T. Aste, R. N. Mantegna, Physica A 339 (2004) 181

Metric distance Correlations Three axioms:if and only if i=j1) 2) 3) J. C. Gower, Biometrika 53 (1966) ; R. N. Mantegna, Eur. Phys. J. B (1999) T 1 and T 2 delimit the range of t is the average over time of Δf i (t) Metric graphs

Extending the MST How to construct a graph richer of links but preserving the same hierarchical structure? R. N. Mantegna, Hierarchical structure in financial markets, Eur. Phys. J. B (1999) MST retains only (n-1) correlation coefficients from the original n(n-1)/2 MINIMUM SPANNING TREE (MST) Eurodollars34 US Interest Rates

Graph g=0 embedded on a sphere

In practice, the magnitudes of the elastic moduli are tuned to ensure convergence to a final configuration with all edges of length equal to d i,j and angles as nearly equal as possible. Network relaxation procedure Vertices i,j,k placed at random in Cartesian space

Eurodollars 34 US Interest Rates Hierarchy

3) CLUSTERING Ultra-metric distance between two elements i,j belonging to two different clusters is the maximum metric distance between all couples of elements in the two clusters. Ultra-Metric distance A Cluster is a set of elements at distances d i,j smaller than a given threshold Disjoined clusters have some elements which are at distances larger than the threshold.

Three main clusters: 1) < 1 year 2) 1-2 years 3) > 2 years Eurodollar interest rates Six main clusters and Three isolated elements

< 1 year years > 2 years 1 month months (no Tr.) months (Tr.) y. > 3 years TBA 3-6 m. FED CM SLB T. Di Matteo, T. Aste, S. T. Hyde and S. Ramsden, Interest rates hierarchical structure, Physica A 355 (2005) CP3, CP6, FP3, FP6, BA3, BA6, CD3, CD6, ED3M, ED6MTC3M, TC6M, TBA3M, TBA6M, TBS3M, TBS6M

M. Tumminello, T. Aste, T. Di Matteo and R. N. Mantegna, A tool for filtering information in complex systems, Proceedings of the National Academy of Sciences of the United States of America Vol. 102, Num. 30 (2005) stocks in the USA equity markets Basic Materials (B) (Pink) Utilities (U) (Yellow) Financial (F) (Cyan) Consumer Non Cyclical (C) (Purple) Consumer Cyclical (CC) (Orange) Capital Goods (CG) (Magenta) Healthcare (H) (Brown) Services (S) (Red) Technology (T) (Green) Conglomerates (CO) (Gray) Energy (E) (Blue) Transportation (TR) (White)

Graph richer of links but preserving the MST hierarchical structure (n-1) 3(n-2) BAC JPMMER MOB XON CHVARC A clique of r elements (r-clique) is a complete subgraph that links all r elements292 = 3n = n - 3 Such loops and cliques have important and significant relations with the market structure and properties

4-cliques structure 31 cliques are composed by stocks belonging to the same economic sector 22 are composed by 3 stocks belonging to the same sector 37 have 2 stocks from the same sector 7 have stocks all from different sectors

M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) most capitalized stocks traded at the NYSE January 2001 – December 2003 Return time series sampled at different time horizons: 5, 15, 30, 65, 130, 195 and 390 min 1 trading day Nature and properties of the PMFG associated to a given financial portfolio as a function of the time horizon used to record stock return time series

5 min time horizon Merrill Lynch co inc (MER) Suntrust banks inc (STI) PPG industries inc (PPG) Eaton corp (ETN) Jefferson-Pilot corp (JP) General Electric (GE) Wal-Mart stores inc (WMT) Basic Materials (violet, 24 stocks), Consumer Cyclical (tan, 22 stocks), Consumer Non Cyclical (yellow, 25 stocks), Energy (blue, 17 stocks), Services (cyan, 69 stocks), Financial (green, 53 stocks), Healthcare (gray, 19 stocks), Technology (red, 34 stocks), Utilities (magenta, 12 stocks), Transportation (brown, 5 stocks), Conglomerates (orange, 8 stocks) and Capital Goods (light green, 12 stocks)

1 day time horizon Merrill Lynch co inc (MER) General Electric (GE) Eaton corp (ETN) PPG industries inc (PPG) Suntrust banks inc (STI) Wal-Mart stores inc (WMT) Jefferson-Pilot corp (JP) Basic Materials (violet, 24 stocks), Consumer Cyclical (tan, 22 stocks), Consumer Non Cyclical (yellow, 25 stocks), Energy (blue, 17 stocks), Services (cyan, 69 stocks), Financial (green, 53 stocks), Healthcare (gray, 19 stocks), Technology (red, 34 stocks), Utilities (magenta, 12 stocks), Transportation (brown, 5 stocks), Conglomerates (orange, 8 stocks) and Capital Goods (light green, 12 stocks)

M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) min time horizon 1 day time horizon

Topological properties Shortest path s(i,j) minimum number of edges crossed by connecting vertices i and j in the graph Betweenness btw(i) number of shortest paths traversing the vertex i Degree k(i) number of edges connected to the vertex i Connection strength ratio between the number of cliques of 3 or 4 elements present among n s stocks belonging to a given set and a normalizing quantity n s – 3 for 4-cliques and 3 n s – 8 for 3-cliques

M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) Average length of shortest path as function of the sampling time horizon of return 195 min

M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) Betweenness of GE and PPG evaluated in the PMFG as function of the time horizon

M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) Degree of GE and PPG evaluated in the PMFG as function of the time horizon

The effect of GE at short time horizons strongly intervenes in the connection between different branches (sectors) of the PMFG whereas at longer time horizon connection between sectors are more complex and the central role of GE progressively disappears GE hub for the whole market at short time horizons its relevance decreases according to the structuring of the market into sectors observed at long time horizon PPG hub for its own economic sector (Basic Materials) it is a local hub both at short and long time horizons sector of basic materials is formed already at short time horizons

Connection strength evaluated by the number of intra-sector 3-cliques (C 3 )

Conglomerates and capital goods Energy, financial and utilities the connection strength is very close to one already at the shortest time horizon. This behavior indicates that the sectors are well defined and driven by the same factors down to a very short time horizon. Consumer cyclical, healthcare and services clearly showing that the market needs a finite time to produce a profile of correlation compatible with the sector classification. Value smaller than 1 at longer time horizons. Basic materials, consumer non cyclical, and technology sectors show an intermediate behavior characterized by a non marked time dependence and moderately low values of the overall connection strength.

Sub-sectors All the considered sub-sectors show a connection strength greater or at most equal to the connection strength of the economic sector they belong to. They are significantly intra-connected before or at most at the same time horizon as the corresponding economic sector.

300 most capitalized stocks traded at the NYSE January 2001 – December 2003 Nature and properties of the MST and PMFG at different time series windows: 1, 2, 3, 4, 6, 12 months moving through the time series Booms Crashes 11/9/200119/7/20029/10/2002

1 month 2 months 3 months 6 months 4 months 12 months

Average distance for 1 month Complete graph Planar graph MST

1 month 2 months 3 months 6 months 4 months 12 months Complete graph

1 month 2 months 6 months 4 months 12 months 3 months Planar graph

1 month 2 months 3 months 6 months 4 months 12 months MST

Persistence of the structure MSTPlanar T1 Planar

Characterization and Visualization of Complex systems by means of Hyperbolic graphs A general tool for Information Filtering Measure of complexity looking at the amount of information necessary to describe the system Efficient in filtering relevant information about the clustering of the system and its hierarchical structure Generate networks with the same hierarchical structure of the MST Triangular loops and 4 element cliques have important and significant relations with the market structure and properties Triangular loops and 4 element cliques have important and significant relations with the market structure and properties The market is progressively structured as a function of the time horizon The market is progressively structured as a function of the time horizon The market structuring occurs by first connecting stocks belonging to the same sub-sector and then connecting stocks belonging to the same economic sector The market structuring occurs by first connecting stocks belonging to the same sub-sector and then connecting stocks belonging to the same economic sector

Under investigation Shortest path DegreeBetweenness Different Sectors Different filtered graphs Effect of g on the information filtering Dynamical graphs and elementary moves